zbMATH — the first resource for mathematics

Efficient quadrature-free high-order spectral volume method on unstructured grids: theory and 2D implementation. (English) Zbl 1134.65070
Summary: An efficient implementation of the high-order spectral volume (SV) method is presented for multi-dimensional conservation laws on unstructured grids. In the SV method, each simplex cell is called a spectral volume, and the SV is further subdivided into polygonal (2D), or polyhedral (3D) control volumes (CVs) to support high-order data reconstructions.
In the traditional implementation, Gauss quadrature formulas are used to approximate the flux integrals on all faces. In the new approach, a nodal set is selected and used to reconstruct a high-order polynomial approximation for the flux vector, and then the flux integrals on the internal faces are computed analytically, without the need for Gauss quadrature formulas. This gives a significant advantage over the traditional SV method in efficiency and ease of implementation.
For SV interfaces, a quadrature-free approach is compared with the Gauss quadrature approach to further evaluate the accuracy and efficiency. A simplified treatment of curved boundaries is also presented that avoids the need to store a separate reconstruction for each boundary cell. Fundamental properties of the new SV implementation are studied and high-order accuracy is demonstrated for linear and non-linear advection equations, and the Euler equations. Several well known inviscid flow test cases are utilized to show the effectiveness of the simplified curved boundary representation.

MSC:
 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 35L65 Hyperbolic conservation laws 35L45 Initial value problems for first-order hyperbolic systems
Software:
HLLE; Mathematica
Full Text:
References:
 [1] Abgrall, R., On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation, J. comput. phys., 114, 45-58, (1994) · Zbl 0822.65062 [2] Atkins, H.L.; Shu, Chi-Wang, Quadrature-free implementation of the discontinuous Galerkin method for hyperbolic equations, Aiaa j., 96, 1683, (1996) [3] T.J. Barth, P.O. Frederickson, High-order solution of the Euler equations on unstructured grids using quadratic reconstruction, AIAA Paper No. 90-0013, 1990. [4] Bassi, F.; Rebay, S., High-order accurate discontinuous finite element solution of the 2D Euler equations, J. comput. phys., 138, 251-285, (1997) · Zbl 0902.76056 [5] Batten, P.; Clarke, N.; Lambert, C.; Causon, D.M., On the choice of wavespeeds for the HLLC Riemann solver, SIAM J. sci. comput., 18, 6, 1553-1570, (1997) · Zbl 0992.65088 [6] Chen, Q.Y., Partitions for spectral (finite) volume reconstruction in the tetrahedron, SIAM J. sci. comput., (2005) [7] Cockburn, B.; Shu, C.-W., TVB runge – kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Math. comput., 52, 411-435, (1989) · Zbl 0662.65083 [8] Cockburn, B.; Lin, S.-Y.; Shu, C.-W., TVB runge – kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems, J. comput. phys., 84, 90-113, (1989) · Zbl 0677.65093 [9] Cockburn, B.; Hou, S.; Shu, C.-W., TVB runge – kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Math. comput., 54, 545-581, (1990) · Zbl 0695.65066 [10] Cockburn, B.; Shu, C.-W., The runge – kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. comput. phys., 141, 199-224, (1998) · Zbl 0920.65059 [11] M. Delanaye, Yen Liu, Quadratic reconstruction finite volume schemes on 3D arbitrary unstructured polyhedral grids, AIAA Paper No. 99-3259-CP, 1999. [12] Godunov, S.K., A finite-difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics, Mat. sb., 47, 271, (1959) · Zbl 0171.46204 [13] Gottlieb, S.; Shu, C.-W.; Tadmor, E., Strong stability-preserving high-order time discretization methods, SIAM rev., 43, 1, 89-112, (2001) · Zbl 0967.65098 [14] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S., Uniformly high order essentially non-oscillatory schemes III, J. comput. phys., 71, 231, (1987) · Zbl 0652.65067 [15] Harten, A.; Lax, P.D.; Van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM rev., 25, 35-61, (1983) · Zbl 0565.65051 [16] Hesthaven, J.S., From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex, SIAM J. numer. anal., 35, 2, 655-676, (1998) · Zbl 0933.41004 [17] Hesthaven, J.S.; Teng, C.H., Stable spectral methods on tetrahedral elements, SIAM J. sci. comput., 21, 6, 2352-2380, (2000) · Zbl 0959.65112 [18] Hu, C.; Shu, C.-W., Weighted essentially non-oscillatory schemes on triangular meshes, J. comput. phys., 150, 97-127, (1999) · Zbl 0926.65090 [19] Krivodonova, L.; Berger, M., High-order accurate implementation of solid wall boundary conditions in curved geometries, J. comput. phys., 211, 492-512, (2006) · Zbl 1138.76403 [20] Liu, Y.; Vinokur, M.; Wang, Z.J., Spectral (finite) volume method for conservation laws on unstructured grids V: extension to three-dimensional systems, J. comput. phys., 212, 454-472, (2006) · Zbl 1085.65099 [21] H. Luo, J. Baum, R. Löhner, On the computation of steady-state compressible flows using a discontinuous Galerkin method, presented at the Fourth International Conference on Computational Fluid Dynamics, July 10-14, Ghent, Belgium, 2006. [22] Roe, P.L., Approximate Riemann solvers, parameter vectors and difference schemes, J. comput. phys., 43, 357-372, (1981) · Zbl 0474.65066 [23] Rusanov, V.V., Calculation of interaction of non-steady shock waves with obstacles, J. comput. math. phys. USSR, 1, 267-279, (1961) [24] Sun, Y.; Wang, Z.J.; Liu, Y., High-order multidomain spectral difference method for the navier – stokes equations on unstructured hexahedral grids, Commun. comput. phys., 2, 310-333, (2007) · Zbl 1164.76360 [25] Sun, Y.; Wang, Z.J.; Liu, Y., Spectral (finite) volume method for conservation laws on unstructured grids VI: extension to viscous flow, J. comput. phys., 215, 41-58, (2006) · Zbl 1140.76381 [26] Toro, E.F.; Spruce, M.; Speares, W., Restoration of the contact surface in the HLL Riemann solver, Shock waves, 4, 25-34, (1994) · Zbl 0811.76053 [27] Tu, S.; Aliabadi, S., A slope limiting procedure in discontinuous Galerkin finite element method for gasdynamics applications, Int. J. numer. anal. mod., 2, 2, 163-178, (2005) · Zbl 1151.76530 [28] van Leer, B., Towards the ultimate conservative difference scheme V. A second-order sequel to godunov’s method, J. comput. phys., 32, 101-136, (1979) · Zbl 1364.65223 [29] van Leer, B., Upwind and high-resolution methods for compressible flow: from donor cell to residual-distribution schemes, Commun. comput. phys., 1, 192-206, (2006) · Zbl 1114.76049 [30] Wang, Z.J., Spectral (finite) volume method for conservation laws on unstructured grids: basic formulation, J. comput. phys., 178, 210, (2002) · Zbl 0997.65115 [31] Wang, Z.J.; Liu, Y., Spectral (finite) volume method for conservation laws on unstructured grids II: extension to two-dimensional scalar equation, J. comput. phys., 179, 665, (2002) · Zbl 1006.65113 [32] Wang, Z.J.; Liu, Y., Spectral (finite) volume method for conservation laws on unstructured grids III: extension to one-dimensional systems, J. sci. comput., 20, 137, (2004) · Zbl 1097.65100 [33] Wang, Z.J.; Liu, Y., Spectral (finite) volume method for conservation laws on unstructured grids IV: extension to two-dimensional Euler equations, J. comput. phys., 194, 716, (2004) · Zbl 1039.65072 [34] Wang, Z.J.; Liu, Y., Extension of the spectral volume method to high-order boundary representation, J. comput. phys., 211, 154-178, (2006) · Zbl 1161.76536 [35] Wolfram, S., Mathematica book, (1999), Wolfram Media and Cambridge University Press New York · Zbl 0924.65002 [36] Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. comput. phys., 54, 115-173, (1984) · Zbl 0573.76057 [37] Xing, Y.; Shu, C.W., A new approach of high order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms, Commun. comput. phys., 1, 100-134, (2006) · Zbl 1115.65096 [38] Zalesak, S.T., Fully multidimensional flux-corrected transport algorithms for fluids, J. comput. phys., 32, 335-362, (1979) · Zbl 0416.76002 [39] Zhang, M.; Shu, C.W., An analysis and a comparison between the discontinuous Galerkin method and the spectral finite volume methods, Comput. fluids, 34, 4-5, 581-592, (2005) · Zbl 1138.76391
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.